Oscillators are ubiquitous in physical systems, especially electronic and optical systems, where oscillators are utilized in phase locked loops, voltage controlled oscillators, microprocessors and transceivers. For example, in radio frequency (RF) communication systems, oscillators are used for frequency translation of information signals and for channel selection. Oscillators are also present in digital electronic systems that require a time reference (or clock signal) to synchronize operations.
Noise is a significant problem with oscillators, because introducing even small noise into an oscillator leads to dramatic changes in its frequency spectrum and timing properties. This phenomenon, peculiar to oscillators, is known as phase noise or timing jitter. A perfect oscillator would have localized tones at discrete frequencies (fundamental frequency, f.sub.0, and harmonics, 2f.sub.0 and 3f.sub.0), as shown in FIG. 1A, but any corrupting noise spreads these perfect tones, resulting in high power levels at neighboring frequencies, as shown in FIG. 1B. Spreading is the major contributor to undesired phenomena, such as interchannel interference, leading to increased bit-error-rates (BER) in RF communication systems.
Another manifestation of the same phenomenon is timing jitter. Timing jitter is important in clocked and sampled-data systems. The uncertainties in switching instants caused by noise lead to synchronization problems. Thus, characterizing how noise affects oscillators is crucial for practical applications. The problem, however, is challenging since oscillators constitute a special class among noisy physical systems. The autonomous nature of oscillators makes them unique in their response to perturbations.
Considerable effort has been expended over the years in understanding phase noise and in developing analytical, computational and experimental techniques for its characterization. For a brief review of such analytical, computational and experimental techniques, see, for example, W. P. Robins, Phase Noise in Signal Sources, Peter Peregrinus (1991); B. Razavi, Analysis, Modeling and Simulation of Phase Noise in Monothilic Voltage-Controlled Oscillators, Proc. IEEE Custom Integrated Circuits Conference (May 1995), each incorporated by reference herein.
In order to compensate for phase noise and timing jitter, the designers of oscillators and devices incorporating oscillators utilize existing commercial tools, such as the tools commercially available from Cadence Design Systems and Hewlett Packard's EESOF, to analyze the single-side band phase noise spectrum as a function of the offset from the fundamental frequency. A typical single-side band phase noise spectrum is shown in FIG. 2. Specific single-side band phase noise values are specified for a given device at various frequencies, in a known manner. Conventional tools for analyzing the single-side band phase noise spectrum typically assume the peak power spectral density (PSD) 150, shown in FIG. 1B, is infinity. Thus, the single-side band phase noise values 210 generated by conventional tools likewise go to infinity, as shown in FIG. 2. It has been found, however, that the estimate of the single-side band phase noise actually has a shape as shown by the curve 220. As a result, if the device being designed is analyzed at a frequency below a given offset frequency, f.sub.off, 230, the oscillator will be over-designed to compensate for the overestimated single-side band phase noise.
Despite the importance of the phase noise problem and the large number of publications on the subject, a consistent and general treatment, and computational techniques based on a sound theory, are still lacking. Thus, a need exists for a method and apparatus using efficient numerical methods for the characterization of phase noise. A further need exists for a technique that is applicable to any oscillatory system, including electrical systems, such as resonant, ring and relaxation oscillators, and other systems, such as gravitational, optical, mechanical and biological oscillators. Yet another need exists for a tool that estimates the actual power spectral density (PSD) of a given oscillator and accurately characterizes the oscillator's single-side band phase noise spectrum.